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The empirical process of a short-range dependent stationary sequence under Gaussian subordination
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  • Published: March 1996

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

  • Sándor Csörgó1 &
  • Jan Mielniczuk2 

Probability Theory and Related Fields volume 104, pages 15–25 (1996)Cite this article

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  • 33 Citations

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Summary

Consider the stationary sequenceX 1=G(Z 1),X 2=G(Z 2),..., whereG(·) is an arbitrary Borel function andZ 1,Z 2,... is a mean-zero stationary Gaussian sequence with covariance functionr(k)=E(Z 1 Z k+1) satisfyingr(0)=1 and ∑ ∞ k=1 |r(k)|m < ∞, where, withI{·} denoting the indicator function andF(·) the continuous marginal distribution function of the sequence {X n }, the integerm is the Hermite rank of the family {I{G(·)≦ x} −F(x):x∈R}. LetF n (·) be the empirical distribution function ofX 1,...,X n . We prove that, asn→∞, the empirical processn 1/2{F n (·)-F(·)} converges in distribution to a Gaussian process in the spaceD[−∞,∞].

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Author information

Authors and Affiliations

  1. Department of Statistics, The University of Michigan, 1440 Mason Hall, 48109-1027, Ann Arbor, MI, USA

    Sándor Csörgó

  2. Institute of Computer Science, Polish Academy of Sciences, Ordona 21, PL-01-237, Warsaw, Poland

    Jan Mielniczuk

Authors
  1. Sándor Csörgó
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  2. Jan Mielniczuk
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Additional information

Partially supported by NSF Grant DMS-9208067

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Csörgó, S., Mielniczuk, J. The empirical process of a short-range dependent stationary sequence under Gaussian subordination. Probab. Th. Rel. Fields 104, 15–25 (1996). https://doi.org/10.1007/BF01303800

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  • Received: 01 April 1994

  • Revised: 01 June 1995

  • Issue Date: March 1996

  • DOI: https://doi.org/10.1007/BF01303800

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Mathematics Subject Classification (1991)

  • 60F17
  • 62G30
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